DOCSLIB.ORG

Harmonic Morphisms with Fibers of Dimension One

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Harmonic Morphisms with Fibers of Dimension One COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 8, Number 2, 219-265, 2000 Harmonic morphisms with fibers of dimension one ROBERT L. BRYANT1 The harmonic morphisms (j) : Mn+1 —> Nn are studied using the methods of the moving frame and exterior differential systems and three main results are achieved. The first result is a local structure theorem for such maps in the case that (f) is a submersion, in particular, a normal form is found for all such (j) once the metric on the target manifold N is specified. The second result is a finiteness theorem, which says, in a certain sense, that, when n > 3, the set of harmonic morphisms with a given Riemannian domain (Mn+1, #) is a finite dimensional space. The third result is the explicit classification when n > 3 of all local and global harmonic morphisms with domain (Mn+1,p), a space of constant curvature. 0. Introduction. A smooth map <j) : M —> N between Riemannian manifolds is said to be a harmonic morphism if, for any harmonic function / on any open set V C iV, the pullback / o </> is a harmonic function on (^^(V) C M. By a simple argument (see §1), any non-constant harmonic morphism </>: M —> N between connected Riemannian manifolds must be a submersion away from a set of measure zero in M. Thus, a necessary condition for the existence of a non-constant harmonic map 0 : M —* N is that dimM > dimN. When the dimension of N is 1, so that iV can be regarded, at least locally, as R with its standard metric, a map </> : M -* N is a harmonic morphism if and only if it is a harmonic function in the usual sense. Thus, at least locally, there are many harmonic morphisms from M to N. 1This research was begun during a visit to IMPA in Rio de Janeiro in July 1996 and was inspired by questions raised during the International Conference on Differential Geometry held at IMPA during that month. The article was written during a visit to the Institute for Advanced Study in Princeton. The author would like to thank IMPA and the IAS for their hospitality and also to acknowledge support from the National Science Foundation through grant DMS-9505125. 219 220 Robert Bryant However, when the dimension of N is greater than 1, the condition of being a harmonic morphism turns out to be much more restrictive, being essentially equivalent to an overdetermined system of PDE for the map </>. Thus, for generic Riemannian metrics on M and AT, one does not expect there to be any harmonic morphisms, even locally. Moreover, in the case that there do exist harmonic morphisms (j) : M —> N for given M and AT, one expects the analysis of the overdetermined system that describes them to involve integrability conditions and other features of overdetermined sys- tems. When both M and N have dimension 2, a harmonic morphism is sim- ply a branched conformal mapping between Riemann surfaces and these are studied by classical methods of complex analysis and Riemann surface theory. When both manifolds have the same dimension n > 2, a non-constant harmonic morphism is a local homothety, i.e., up to a constant scale factor, (j) is a local isometry. Thus, the interesting cases are when dimM > dim AT > 2. This article concerns the case when dimM = dimN + 1, i.e., when the dimension of the generic fiber of (/> is 1. It contains three main results. The first, Theorem 1, is a local structure theorem for harmonic mor- phisms whose fibers are curves. This result describes the possible Rieman- nian metrics g that can be defined on the domain M of a smooth map- ping cj) : M —► N where AT is a smooth manifold endowed with a fixed Riemannian metric h so that cj) : (M, g) —► (AT, h) will be a harmonic mor- phism. The second result, Theorem 2, is a general finiteness theorem for har- monic morphisms of corank one with a given Riemannian domain (Mn+1, g) where n > 3. This result shows that the set of such harmonic morphisms is, in a certain sense, finite dimensional. This result is in marked contrast to the case n = 2, which has already been analyzed by Baird and Wood with the result that the locally defined harmonic morphisms with a given Rieman- nian domain (M3,g) of constant sectional curvature depend on arbitrary functions (in the sense of exterior differential systems). The third result, Theorem 3, is a classification of the harmonic mor- phisms of corank one whose domain (Mn+1,^) is a simply-connected, com- plete Riemannian manifold of constant curvature and dimension n+1 > 4. It will be shown that there are exactly two types of such harmonic morphisms. The first type can be thought of as a sort of metric quotient and is described as follows: Let X be a Killing vector field on M with zero lo- cus Z C M and suppose that the space N of integral curves of X in M\Z can Harmonic morphisms with fibers of dimension one 221 be given the structure of a smooth n-manifold in such a way that the quotient map (f): M\Z —>iVisa smooth submersion. Then there exists a metric h on N, unique up to a constant scale factor, so that </> : (M \Z,g) —» (iV, h) is a harmonic morphism. (Sometimes this map can be extended across Z as well after suitably extending AT, see §3.3.) The second type is described as follows: Let iV C M be a totally umbilic hypersurface, endowed with a constant multiple of the induced metric, denoted h. Let P C M be the focal set of AT, which consists of at most two points. There is a canonical retraction (j): M\P —> N that retracts M\P back to N along the geodesies normal to iV. Then ^ is a harmonic morphism. The examples of this kind had already appeared in the work of Gudmundsson [Gul]. The methods used are those of exterior differential systems and the mov- ing frame, both of which are well-adapted to the study of overdetermined systems of PDE. Acknowledgments. It is a pleasure to thank John Wood, whose questions inspired this article and whose comments and and guide to the literature on harmonic morphisms were invaluable. 1. Harmonic morphisms via moving frames. This section is a self-contained treatment by moving frame calculations of the the basic structure theory of harmonic morphisms. It is intended to be readable by those familiar with either moving frame calculations or the fundamentals of harmonic morphisms. Its main purpose is to fix notation and to serve as a reference for the proofs in the later sections, which employ the moving frame. Such a ref- erence is probably needed, as it appears that most of the current workers on harmonic morphisms do not use moving frames and so the translation of known results into this language may be helpful. For more background on the method of the moving frame, see [Sp]. Most of the results about harmonic maps and morphisms to be derived in this section can be found in the standard references on the subject, such as [ELI], [EL2], or [W2]. 1.1. Moving frame computations for harmonic morphisms. Let M and N be Riemannian manifolds of dimensions m and n, respectively. For simplicity, I assume that both M and N are connected throughout this article. The summation convention will be used extensively, with the 222 Robert Bryant understood ranges 1 < a, fe, c < m, 1 < iijjk < n. 1.1.1. Coframings, connection forms, and structure equations. Let g be the metric on M and h be the metric on N. Let U C M and V C N be open sets with trivial tangent bundles. Then there exist smooth cofram- ings u = (a;i,...,a;m) and 77 = (771,... ,ryn) on U and V respectively, so that h\v = r)i + -- + vl = Vi2- Corresponding to the chosen coframings on the respective open sets, there exist unique 1-forms u;a& = — c^ and rjij = —rjji that represent the Levi- Civita connections of the respective metrics and that are characterized by the structure equations (2) duja = -u>ab A ujb , drji = —rjij A rjj . 1.1.2. Mappings and pullbacks. Now suppose that (/> : M —» N is a smooth map and that U and V have been chosen so that U C /_1(y). Then there exist unique functions fia on £/ so that (3) <f)*(Vi) = fiaUa- Because the chosen coframings are orthonormal, the energy density of the map cj) on U is given by E(<t>)\U = /ia/ia |^1 A . AU;n| This density is globally defined, independent of the local choice of u or TJ. When M is compact, integration of this density yields a functional called the energy £ : C00(M, AT) -> R, namely Adopt the convention that, for any differential form ip on V, its <^- pullback </)*('0) on [/ is denoted by an overbar, i.e., ^ = (j)*(ip). Since Harmonic morphisms with fibers of dimension one 223 the map </> will be fixed in this discussion, this should cause no confusion. Thus, (3) becomes (S7) rfi = fia Va . (The reader of other sources on moving frame calculations should be aware that many authors simply drop the pullback notation entirely, writing (3) in the even simpler form rji = fia^a- This has caused considerable confusion in some cases, a confusion I hope to avoid.) Taking the exterior derivative of (3) and using the structure equations (2) yields (dfia - fib Vba + fja Wj) A ^a = 0. By Cartan's Lemma, there exist unique functions /^ = /^a on U so that (4) dfia = fib Uba - fja TJij + fiab ^b - The tension field offionU is the tensor field 7"(0) = fiaa ei0(f> (where (ei,..
Load more

Recommended publications
  • Harmonic Morphisms and Stable Minimal Submanifolds
    Kangweon-Kyungki Math. Jour. 14 (2006), No. 1, pp. 13{33 HARMONIC MORPHISMS AND STABLE MINIMAL SUBMANIFOLDS Gundon Choi and Gabjin Yun¤y Abstract. In this article, we study the relations of horizontally conformal maps and harmonic morphisms with the stability of min- imal ¯bers. Let ' :(M n; g) ! (N m; h) be a horizontally conformal submersion. There is a tensor T measuring minimality or totally geodesics of ¯bers of '. We prove that if T is parallel and the horizontal distribution is integrable, then any minimal ¯ber of ' is volume-stable. As a corollary, we obtain that any ¯ber of a sub- mersive harmonic morphism whose ¯bers are totally geodesics and the horizontal distribution is integrable is volume-stable. As a conse- quence, we obtain if ' :(M n; g) ! (N 2; h) is a submersive harmonic morphism of minimal ¯bers from a compact Riemannian manifold M into a surface N, T is parallel and the horizontal distribution is integrable, then ' is energy-stable. 1. Introduction The theory of harmonic morphisms is one of particularly interesting subclasses of harmonic maps. A harmonic map ' :(M; g) ! (N; h) be- tween Riemannian manifolds is a critical point of the energy functional de¯ned on each compact domain of M. A harmonic morphism between Riemannian manifolds is a map preserving harmonic structure. In other words, a map ' :(M n; g) ! (N m; h) is called a harmonic morphism if for any harmonic function f de¯ned on an open subset V ½ N such Received November 28, 2005. 2000 Mathematics Subject Classi¯cation: 53C43, 58E20.
    [Show full text]
  • HARMONIC MORPHISMS BETWEEN RIEMANNIAN MANIFOLDS by Bent FUGLEDE
    Ann. Inst. Fourier, Grenoble 28, 2 (1978), 107-144, HARMONIC MORPHISMS BETWEEN RIEMANNIAN MANIFOLDS by Bent FUGLEDE Introduction. The harmonic morphisms of a Riemannian manifold, M , into another, N , are the morphisms for the harmonic struc- tures on M and N (the harmonic functions on a Riemannian manifold being those which satisfy the Laplace-Beltrami equation). These morphisms were introduced and studied by Constantinescu and Cornea [4] in the more general frame of harmonic spaces, as a natural generalization of the conformal mappings between Riemann surfaces (1). See also Sibony [14]. The present paper deals with harmonic morphisms between two Riemannian manifolds M and N of arbitrary (not necessarily equal) dimensions. It turns out, however, that if dim M < dim N , the only harmonic morphisms M -> N are the constant mappings. Thus we are left with the case dim M ^ dim N . If dim N = 1 , say N = R , the harmonic morphisms M —> R are nothing but the harmonic functions on M . If dim M = dim N = 2 , so that M and N are Riemann surfaces, then it is known that the harmonic morphisms of (1) We use the term « harmonic morphisms » rather than « harmonic maps » (as they were called in the case of harmonic spaces in [4]) since it is necessary (in the present frame of manifolds) to distinguish between these maps and the much wider class of harmonic maps in the sense of Eells and Sampson [7], which also plays an important role in our discussion. 108 B. FUGLEDE M into N are the same as the conformal mappings f: M -> N (allowing for points where df= 0).
    [Show full text]
  • Harmonic Morphisms Between Riemannian Manifolds Annales De L’Institut Fourier, Tome 28, No 2 (1978), P
    ANNALES DE L’INSTITUT FOURIER BENT FUGLEDE Harmonic morphisms between riemannian manifolds Annales de l’institut Fourier, tome 28, no 2 (1978), p. 107-144 <http://www.numdam.org/item?id=AIF_1978__28_2_107_0> © Annales de l’institut Fourier, 1978, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 28, 2 (1978), 107-144, HARMONIC MORPHISMS BETWEEN RIEMANNIAN MANIFOLDS by Bent FUGLEDE Introduction. The harmonic morphisms of a Riemannian manifold, M , into another, N , are the morphisms for the harmonic struc- tures on M and N (the harmonic functions on a Riemannian manifold being those which satisfy the Laplace-Beltrami equation). These morphisms were introduced and studied by Constantinescu and Cornea [4] in the more general frame of harmonic spaces, as a natural generalization of the conformal mappings between Riemann surfaces (1). See also Sibony [14]. The present paper deals with harmonic morphisms between two Riemannian manifolds M and N of arbitrary (not necessarily equal) dimensions. It turns out, however, that if dim M < dim N , the only harmonic morphisms M -> N are the constant mappings. Thus we are left with the case dim M ^ dim N .
    [Show full text]
  • Polynomial Harmonic Morphisms
    Polynomial Harmonic Morphisms Martin Svensson Examensarb ete p oang Lunds Universitet Novemb er Contents Abstract Acknowledgements Chapter Intro duction Motivation History Chapter Harmonic Maps The Second Fundamental Form Harmonic Maps Harmonic Functions Chapter Harmonic Morphisms Horizontal Conformality Harmonic Morphisms The Existence Problem Polar Sets Chapter Polynomial Harmonic Morphisms Globally Dened Harmonic Morphisms The Classication of Ou Polynomial Harmonic Morphisms of Higher Degree Chapter Applications The Theorems of Eells and Yiu The Symb ol of Harmonic Morphisms Bibliography 1 Abstract The aim of this Masters thesis is to b e the rst survey of known results on p olynomial harmonic morphims b etween Euclidean spaces These were rst studied by Baird in in the early s He obtained several results on the sub ject but left op en the still unsolved classica tion of such maps In the article from Eells and Yiu classied the homogeneous p olynomial harmonic morphisms whose restrictions to spheres are again harmonic morphisms to spheres These are the well known Hopf p olynomials of degree This result revitalized the sub ject and so on thereafter Ou obtained a complete classication of the homogeneous p olynomial harmonic morphisms of degree During the preparation of this thesis a very interesting development has taken place with Abab ou Baird and Brossard writing the article proving that this is still a very active area of research In Chapter we discuss the W eierstrass representation
    [Show full text]
  • Harmonic Morphisms Onto Riemann Surfaces and Generalized Analytic Functions Annales De L’Institut Fourier, Tome 37, No 1 (1987), P
    ANNALES DE L’INSTITUT FOURIER PAUL BAIRD Harmonic morphisms onto Riemann surfaces and generalized analytic functions Annales de l’institut Fourier, tome 37, no 1 (1987), p. 135-173 <http://www.numdam.org/item?id=AIF_1987__37_1_135_0> © Annales de l’institut Fourier, 1987, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 37,1 (1987), 135-173 HARMONIC MORPHISMS ONTO RIEMANN SURFACES AND GENERALIZED ANALYTIC FUNCTIONS by Paul BAIRD Introduction. Let P(w,z) be a polynomial in the complex variables w and z with constant coefficients, having degree n in w. It is well known from the theory of algebraic function that the equation P(w,z) = 0 (1) locally determines n function elements w^ ,w^ , . ,H^ which vary analytically as functions of z. These n function elements determine an algebraic function w together with its associated Riemann surface. This is the starting point for the theory of compact Riemann surfaces [19]. A generalization of these ideas to higher dimensional domains was considered by Jacobi [18]. Let TT (x ,y ,z ,<p) be an analytic function in the real variables x ,y ,z and the complex variable ^, such that TT satisfies the equations (2) = 0.
    [Show full text]
  • Harmonic Morphisms and Submanifolds with ∗ Conformal Second Fundamental Forms
    Glasgow Math. J. 45 (2003) 143–151. C 2003 Glasgow Mathematical Journal Trust. DOI: 10.1017/S001708950200109X. Printed in the United Kingdom HARMONIC MORPHISMS AND SUBMANIFOLDS WITH ∗ CONFORMAL SECOND FUNDAMENTAL FORMS XIAOHUAN MO LMAM School of Mathematical Sciences, Peking University, Beijing 100871, China e-mail: [email protected] (Received 19 November, 2001; accepted 30 May, 2002) Abstract. We show that surfaces such that the natural projections of the unit normal bundles are harmonic morphisms are composed of minimal points and totally umbilical points. As its application, we find a harmonic map from the torus to the complex quadric in CP3 such that the projection map of the associated sphere bundle constructed by Gudmundsson is not a harmonic morphism. This contrasts sharply with the situation for holomorphic maps. We also establish sufficient conditions for reducing the codimension of an isometric immersion with conformal second fundamental form. 2000 Mathematics Subject Classification. 53C12, 53C43, 58E20. 1. Introduction. Aclass of harmonic maps with a strong geometric flavour is the class of harmonic morphisms. Harmonic morphisms between Riemannian manifolds are solutions to an over-determined non-linear system of partial differential equtions. For this reason they have no general existence theory, not even locally. Forany holomorphic map ξ : M → Qn from a Riemannian surface M to an n- dimensional complex quadric Qn, Gudmundsson constructed in [9](cf.(4.1)) a sphere bundle Mξ over M with projection map that is a harmonic morphism with totally geodesic fibres; this was later generalized by Aprodu and Aprodu ([1]). Eells and Sampson proved in [6]that any holomorphic map between Kaehler manifolds is harmonic.
    [Show full text]
  • Harmonic Morphisms on Riemannian Manifolds
    Harmonic Morphisms on Riemannian Manifolds Andreas Quist June 2019 Abstract The goal of this paper is to define and charaterize harmonic mor- phisms between Riemannian manifolds, then build up machinery for utilizing them. Much of the paper will follow the works of Bent Fu- glede, who pioneered much of the theory of harmonic morphisms in his seminal paper 40 years ago. Later we will take an algebraic approach to the study of harmonic morphisms, building up some categorical machin- ery for future use. We conclude with an example of utilizing harmonic morphisms by applying them to solving the Dirichlet problem on Rie- mannian manifolds. Section 1 is dedicated to introductory material, 2 follows the works of Fuglede, and 3 and 4 contain interesting findings of my own. Contents 1 Introduction 2 1.1 Riemannian Manifolds . .2 1.2 Sheafs . .5 2 Harmonic Morphisms 7 2.1 Preliminaries for Characterization . .8 2.2 Characterization of Harmonic Morphisms . 10 3 The Categories BHarm and BSheaf 12 4 Harmonic Morphisms and the Dirichlet Problem 13 4.1 Solutions to the Dirichlet Problem on Riemannian Manifolds . 14 1 1 Introduction This paper makes use of some concepts from elementary category theory. The basic definitions of categories, morphisms, and functors are assumed to be known by the reader beforehand. However, some definitions will be provided in this section. All category theory used in this paper is explained in greater detail in Steve Awodey's book Category Theory [2]. We will utilize the notion of contravariant functors, which play a critical role in the definition of sheaves.
    [Show full text]
  • HARMONIC MAPS Introduction 1 Harmonic Functions on Euclidean
    Article for the ‘Handbook of Global Analysis’, Elsevier (2007, to appear) HARMONIC MAPS Fred´ eric´ Helein´ and John C. Wood Introduction The subject of harmonic maps is vast and has found many applications, and it would require a very long book to cover all aspects, even superficially. Hence, we have made a choice; in particular, highlighting the key questions of existence, uniqueness and regularity of harmonic maps between given manifolds. Thus we shall survey some of the main methods of global analysis for answering these questions. We first consider relevant aspects of harmonic functions on Euclidean space; then we give a general introduc- tion to harmonic maps. The core of our work is in Chapters 3–6 where we present the analytical methods. We round of the article by describing how twistor theory and integrable systems can be used to construct many more harmonic maps. On the way, we mention harmonic morphisms: maps between Riemannian manifolds which pre- serve Laplace’s equation; these turn out to be a particular class of harmonic maps and exhibit some properties dual to those of harmonic maps. More information on harmonic maps can be found in the following articles and books; for generalities: [61, 62, 63, 219], analytical aspects: [21, 88, 103, 118, 131, 133, 135, 189, 204, 194], integrable systems methods: [73, 94, 117], applications to complex and Kahler¨ geometry: [63, 135], harmonic morphisms: [7], and other topics: [64, 231]. 1 Harmonic functions on Euclidean spaces Harmonic functions on an open domain Ω of Rm are solutions of the Laplace equation ∂2 ∂2 ∆f = 0, where ∆ := + ··· + (x1, ··· , xm) ∈ Ω.
    [Show full text]
  • Twistorial Constructions of Harmonic Morphisms and Jacobi Fields
    Twistorial constructions of harmonic morphisms and Jacobi fields Bruno Manuel Ascenso da Silva Simoes˜ Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Pure Mathematics September 2007 arXiv:1003.5644v1 [math.DG] 29 Mar 2010 The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. To my wife Joana It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. Gauss (1808) O Binomio´ de Newton et´ ao˜ belo como a Venus´ de Milo. O queha´ e´ pouca gente para dar por isso. Alvaro´ de Campos (Fernando Pessoa) (1928) v Acknowledgements First of all, I am very grateful to my supervisor Professor John C. Wood for his infinite patience during these years. Without his compelling enthusiasm for my ideas throughout my stay in Leeds, this work would have never come to fruition. To my colleague and friend Martin Svensson, I would like to dedicate as many lines in this work as the hours we spent debating ideas, mathematical and otherwise. A special thanks to Professor Maria Jo˜ao Ferreira for her interest in my work and for all the support she has always given me. I thank all my friends in Leeds without whom I would not have enjoyed my stay half as much.
    [Show full text]
  • Arxiv:2101.11879V1 [Math.DG]
    PROPER BIHARMONIC MAPS AND (2, 1)-HARMONIC MORPHISMS FROM SOME WILD GEOMETRIES ELSA GHANDOUR AND SIGMUNDUR GUDMUNDSSON Abstract. In this work we construct a variety of new complex-valued proper biharmonic maps and (2, 1)-harmonic morphisms on Riemannian manifolds with non-trivial geometry. These are solutions to a non-linear system of partial differential equations depending on the geometric data of the manifolds involved. 1. Introduction The concept of a harmonic morphism φ : (M, g) → (N, h), between Rie- mannian manifolds, was introduced by Fuglede and Ishihara in the late 1970s independently, see [2] and [6]. These are maps pulling back local real-valued harmonic functions on N to harmonic functions on M. These objects have an interesting connection to the geometry of the manifolds in- volved and have lead to vibrant research activities, as can be traced in the excellent work [1], by Baird and Wood, and the regularly updated online bibliography [5], maintained by the second author. Recently, the notion was generalised to (p,q)-harmonic morphisms, pulling back real-valued q-harmonic functions on N to p-harmonic functions on M, see [3]. The case of (2, 1) had earlier been studied in [4] under the name generalised harmonic morphisms. In [3], the authors characterise complex- valued (p,q)-harmonic morphisms φ : (M, g) → C in terms of a heavily non-linear system of partial differential equations. They also provide meth- ods for producing explicit solutions in the case when the domain (M, g) is the m-dimensional Euclidean space. arXiv:2101.11879v1 [math.DG] 28 Jan 2021 The principal aim of this work is to extend the study to complex-valued (2, 1)-harmonic morphisms from Riemannian manifolds (M, g).
    [Show full text]
  • The Structure of Harmonic Morphisms with Totally
    Available at: http://www.ictp.trieste.it/~pub off IC/99/90 United Nations Educational Scienti c and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE STRUCTURE OF HARMONIC MORPHISMS WITH TOTALLY GEODESIC FIBRES 1 M.T. Mustafa Faculty of Engineering Sciences, GIK Institute of Enginneering Sciences and Technology, Topi-23460, N.W.F.P., Pakistan and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract The structure of lo cal and global harmonic morphisms b etween Riemannian manifolds, with totally geo desic bres, is investigated. It is shown that non-p ositive curvature obstructs the existence of global harmonic morphisms with totally geo desic bres and the only such maps from compact Riemannian manifolds of non-p ositive curvature are, up to a homothety, totally geo desic Riemannian submersions. Similar results are obtained for lo cal harmonic morphisms with totally geo desic bres from op en subsets of non-negatively curved compact and non-compact manifolds. During the course, of this work we prove the non-existence of submersive harmonic morphisms with totally geo desic bres from some imp ortant domains, for instance from compact lo cally symmetric spaces of non-compact typ e and op en subsets of symmetric spaces of compact typ e. MIRAMARE { TRIESTE August 1999 1 Regular Asso ciate of the Ab dus Salam ICTP. E-mail: [email protected] 2 1. Introduction Harmonic morphisms are maps b etween Riemannian manifolds which preserve germs of har- monic functions, i.e. these lo cally pull back real-valued harmonic functions to real-valued har- monic functions.
    [Show full text]
  • HARMONIC MORPHISMS and BICOMPLEX MANIFOLDS 3 Bicomplex Manifolds
    HARMONIC MORPHISMS AND BICOMPLEX MANIFOLDS PAUL BAIRD AND JOHN C. WOOD* Abstract. We use functions of a bicomplex variable to unify the ex- isting constructions of harmonic morphisms from a 3-dimensional Eu- clidean or pseudo-Euclidean space to a Riemannian or Lorentzian sur- face. This is done by using the notion of complex-harmonic morphism between complex-Riemannian manifolds and showing how these are given by bicomplex-holomorphic functions when the codomain is one-bicomplex dimensional. By taking real slices, we recover well-known compactifica- tions for the three possible real cases. On the way, we discuss some interesting conformal compactifications of complex-Riemannian mani- folds by interpreting them as bicomplex manifolds. 1. Introduction Harmonic morphisms are maps ϕ : M N between Riemannian or semi- → Riemannian manifolds which preserve Laplace’s equation in the sense that, if f : V R is a harmonic function on an open subset of N with ϕ−1V → non-empty, then f ϕ is a harmonic function on ϕ−1V . In the Riemannian ◦ case, they can be characterized as harmonic maps which are horizontally weakly conformal (also called semiconformal), a condition dual to weak conformality. The characterization can be extended to harmonic morphisms between semi-Riemannian manifolds, with the additional feature that fibres can be degenerate. Harmonic morphisms into Riemannian or Lorentzian surfaces are par- ticularly nice as they are conformally invariant in the sense that only the arXiv:0910.1036v2 [math.DG] 11 Mar 2010 conformal equivalence class of the metric on the codomain matters; equiva- lently postcomposition of a harmonic morphism to a surface with a weakly conformal map of surfaces is again a harmonic morphism.
    [Show full text]